Equation (REF ) speaks towards the general hashing bound of the whole family of Pauli channels. Sub-families of Pauli channels of special interest are the ones obtained by \(\mathcal {P}_n\) - and \(\mathcal {C}_1^{\otimes n}\) -twirling the AD channel [1]}, [2]}, [3]}, [4]}. The families of Pauli channels obtained by such operations are denominated the AD Pauli twirl approximated (ADPTA) channel and the AD Clifford twirl approximated (ADCTA) channel, which is a depolarizing channel (since \(p_x=p_y=p_z\) ). The parameters \(p_x,p_y,p_z\)  are themselves functions of the relaxation time, \(T_1\) , due to the fact that they are approximated by the \(T_1\) -dependent AD channel.
